A Note on the Kervaire Invariant

In [5] M. Kervaire defined an invariant for (4fc+2)-dimensional framed manifolds. This invariant depends only on the framed bordism class of the manifold and lies in Z2. W. Browder [2] (see also E. H. Brown [3]) gave a generalisation of the invariant that is defined for any even dimensional manifold with a Wu orientation; in this case it depends only on the Wu bordism class. A framed manifold has a Wu orientation and using his generalisation Browder showed that the Kervaire invariant of M" is zero unless n = 2 — 2 for some r > 1. In this note we reprove the above mentioned result of Browder. We use a consequence, due to NigeJ Ray [7], of the theorem of D. S. Kahn and S. Priddy [4]. This allows us to avoid the computational part of the proof in [2]. Throughout, all homology and cohomology groups have Z2 coefficients and we denote the Eilenberg-Maclane space K(Z2, n) by Kn.